Problem: There exists a constant $c,$ so that among all chords $\overline{AB}$ of the parabola $y = x^2$ passing through $C = (0,c),$
\[t = \frac{1}{AC^2} + \frac{1}{BC^2}\]is a fixed constant.  Find the constant $t.$

[asy]
unitsize(1 cm);

real parab (real x) {
  return(x^2);
}

pair A, B, C;

A = (1.7,parab(1.7));
B = (-1,parab(-1));
C = extension(A,B,(0,0),(0,1));

draw(graph(parab,-2,2));
draw(A--B);
draw((0,0)--(0,4));

dot("$A$", A, E);
dot("$B$", B, SW);
dot("$(0,c)$", C, NW);
[/asy]
Explanation: Let $y = mx + c$ be a line passing through $(0,c).$  Setting $y = x^2,$ we get
\[x^2 = mx + c,\]or $x^2 - mx - c = 0.$  Let $x_1$ and $x_2$ be the roots of this equation.  By Vieta's formulas, $x_1 + x_2 = m$ and $x_1 x_2 = -c.$

Also, $A$ and $B$ are $(x_1,mx_1 + c)$ and $(x_2,mx_2 + c)$ in some order, so
\begin{align*}
\frac{1}{AC^2} + \frac{1}{BC^2} &= \frac{1}{x_1^2 + m^2 x_1^2} + \frac{1}{x_2^2 + m^2 x_2^2} \\
&= \frac{1}{m^2 + 1} \left (\frac{1}{x_1^2} + \frac{1}{x_2^2} \right) \\
&= \frac{1}{m^2 + 1} \cdot \frac{x_1^2 + x_2^2}{x_1^2 x_2^2} \\
&= \frac{1}{m^2 + 1} \cdot \frac{(x_1 + x_2)^2 - 2x_1 x_2}{(x_1 x_2)^2} \\
&= \frac{1}{m^2 + 1} \cdot \frac{m^2 + 2c}{c^2}.
\end{align*}For this expression to be independent of $m,$ we must have $c = \frac{1}{2}.$  Hence, the constant $t$ is $\boxed{4}.$